*3.1. Spectral Local Linearization Scheme*

Let us having a system of differential equations *G* = *<sup>g</sup>*1(ξ), *g*1(ξ), ... , *gp*(ξ)satisfying the system:

$$\mathcal{L}\_{\text{j}} + \overline{\mathcal{N}}\_{\text{j}} = \mathcal{H}\_{\text{j}}, \text{ j} = 1, 2, \dots, p,\tag{18}$$

where *p* describes the number of differential equations, each <sup>H</sup>*j* is a function of ξ[*<sup>A</sup>*, *B*] and L*j*, N *j* are the linear and nonlinear components in the system, respectively.

Usually, the SLLM is an iterative approach to solve the differential equations, starts from an initial approximation *g*0, and then implements the SLLM successively, yielding the new approximations *g*1, *g*2, ..., where *Gt* = *<sup>g</sup>*1,*t*, *g*2,*t*, ... , *gp*,*<sup>t</sup>* for each *t* = 0, 1, 2. When once linearized, the nonlinear components are N 0 *j*.

For this intention, the *j*-th differential Equation (18) after the first *t* + 1 iterations can be express as

$$\left. \mathcal{L}\_{\dot{\jmath}} \right|\_{t+1} + \left. \widetilde{\mathcal{N}}\_{\dot{\jmath}} \right|\_{t+1} = \mathcal{H}\_{\dot{\jmath}}.\tag{19}$$

0

The nonlinear components can be linearized by using Taylor series

$$\left.\overline{\mathbf{N}}\_{\dot{l}}\right|\_{t+1} = \left.\overline{\mathbf{N}}\_{\dot{l}}\right|\_{t} + \nabla \overline{\mathbf{N}}\_{\dot{l}}\Big|\_{t} [V\_{t+1} - V\_{t}]\_{\prime} \tag{20}$$

where *Vt* is an n-tuple of *Gj*,*<sup>t</sup>* and its differentials. Now using Equations (19) and (20) in Equation (18), it becomes

$$\left. \mathcal{L}\_{\dot{j}} \right|\_{t+1} + \nabla \widetilde{\mathbf{N}}\_{\dot{j}} \Big|\_{t} V\_{t+1} = \left. \mathcal{H}\_{\dot{j}} + \nabla \widetilde{\mathbf{N}}\_{\dot{j}} \right|\_{t} V\_{t} - \widetilde{\mathbf{N}}\_{\dot{j}} \Big|\_{t}. \tag{21}$$
